On the Cauchy problem for degenerated difference equations in Banach space (Q2768820)

The authors obtain the sufficient conditions for solvability of the difference equation \(x_{n+1}=x_{n}+f_{n}(x_{n}),\;n\in Z_0^{+},\) with the condition \(x_{k}(x_0)=d\in W\), where \(k\in\{1,2,\ldots\}\) is fixed; \(x_{n}\in W\); \(f_{n}:W\to W,\;\forall n\in Z_0^{+}\); \(W\) is a Banach space; \(Z_0^{+}=\{0,1,2,3,\ldots\}\). Let us suppose that \(x_{k}(x_0)\neq d\), where \(x_{n}(x_0)\) is a solution of the considered equation. The authors prove that if for all \(n\in\{1,2,3,\ldots,k-1\}\) and \(\forall\{x,\bar x\}\subset W\)\ \(\|f_{n}(x)-f_{n}(\bar x)\|\leq K\|x-\bar x\|\), \(K>0,\;2kK<1\), then for all \(x_0\in W\) there exists \(\alpha(x_0)\in W\) such that a solution \(z_{n}(x_0)\) of the disturbed equation \(z_{n+1}=z_{n}+f_{n}(z_{n})+\alpha(x_0),\;n\in Z_0^{+}\) satisfies the condition \(z_{k}(x_0)=d\).